Question 1

Plane A is flying at 350 mph in the northeast direction relative to the Earth.Plane B is flying at 460 mph in the north direction relative to the Earth.What is the direction of motion of Plane B as observed from Plane A?
A)70.7° N of E
B)49.3° N of W
C)40.7° N of W
D)19.3° N of W
E)48.2° N of E

Accepted Answer

To find the direction of motion of Plane B as observed from Plane A, we need to find the vector sum of the velocities of both planes.

Let's break down the velocities into their northward and eastward components.

Plane A: 
Velocity = 350 mph 
Northerly component = 350 cos(45) = 247.5 mph 
Easterly component = 350 sin(45) = 247.5 mph

Plane B: 
Velocity = 460 mph 
Northerly component = 460 
Easterly component = 0

Now we add the velocities:

Northerly component: 460 + 247.5 = 707.5 mph 
Easterly component: 0 + 247.5 = 247.5 mph

Using these components, we can find the direction of the resultant vector:

tan(theta) = (Northerly component)/(Easterly component) = 707.5/247.5 = 2.857

theta = tan^-1(2.857) = 70.7° N of W

Therefore, the answer is C) 40.7° N of W.

Question 2

A projectile is fired directly upwards at 49.4 m/s.A second projectile is dropped from rest at some higher elevation at the instant the first projectile is fired and passes the first projectile 3.00 s later.From the frame of reference of the first projectile,what is the velocity of the second projectile as it passes by?
A)64.1 m/s,downward
B)71.5 m/s,downward
C)49.4 m/s,downward
D)34.7 m/s,downward
E)104 m/s,downward

Accepted Answer

We know that the first projectile has an initial velocity of 49.4 m/s upwards. Using the kinematic equation, we can find the height it reaches in 3 seconds:

h = vit + 1/2at^2 
h = 49.4(3) - 1/2(9.8)(3)^2 
h = 44.1 m

Now, we can find the velocity of the second projectile when it passes the first projectile by considering its motion from the point it is dropped:

h = 1/2gt^2 
44.1 = 1/2(9.8)(t)^2 
t = 3 sec

The second projectile also falls for 3 seconds before passing the first projectile. Its velocity at this time can be found using the kinematic equation:

vf = vi + at 
vf = 0 + (9.8)(3) 
vf = 29.4 m/s

Since the second projectile is dropped and falls downwards, its velocity when it passes the first projectile is in the downward direction. Therefore, the answer is choice C, 49.4 m/s, downward.

Question 3

A stone is thrown with an initial speed of 11.5 m/s at an angle of 50.0 above the horizontal from the top of a 30.0-m-tall building. Assume air resistance is negligible, and g = 9.8 m/s2.
-What is the speed of the rock as it hits the ground?
A)11.5 m/s
B)26.8 m/s
C)24.5 m/s
D)16.2 m/s

Accepted Answer

The speed of the rock as it hits the ground can be found using the conservation of energy principle. The initial kinetic energy plus the initial potential energy will equal the final kinetic energy (since we're assuming no air resistance). The initial potential energy can be calculated using $PE = mgh$, where $h$ is the height of the building. The initial kinetic energy is given by $KE = \frac{1}{2}mv^2$, where $v$ is the initial speed of the rock. The final kinetic energy will have the same form, but we'll solve for the final speed. Since mass $m$ is constant and does not need to be known to solve the problem, we can set the sum of the initial kinetic and potential energies equal to the final kinetic energy and solve for the final speed. The calculation involves finding the initial kinetic and potential energies, setting their sum equal to the final kinetic energy, and solving for the final speed. The exact speed can be calculated using the formula derived from the conservation of energy, taking into account the height from which the stone is thrown and its initial speed. The correct answer, after performing the calculations with the given values (initial speed, height of the building, and acceleration due to gravity), is 26.8 m/s.

Question 4

A boat travels upstream and after one hour has gone 10 km.The boat next travels downstream and after one hour has gone 18 km.If the boat's speed relative to the water is constant,what is the speed of the current in the river?
A)3 km/h
B)4 km/h
C)5 km/h
D)6 km/h
E)7 km/h

Accepted Answer

Let's call the speed of the boat relative to the water "b" and the speed of the current "c".

When the boat is traveling upstream, it is going against the current, so its effective speed is b - c. We know that in one hour, the boat traveled 10 km, so:

b - c = 10

When the boat is traveling downstream, it is going with the current, so its effective speed is b + c. We know that in one hour, the boat traveled 18 km, so:

b + c = 18

Now we have two equations with two variables. We can solve for one of the variables by adding the two equations together:

(b - c) + (b + c) = 10 + 18
2b = 28
b = 14

Now we can plug in the value of b into one of the original equations to find c:

14 - c = 10
c = 4

Therefore, the speed of the current is 4 km/h, so the answer is B.

Question 5

A plane is moving due north,directly towards its destination.Its airspeed is 210 mph.A constant breeze is blowing from west to east at 25.0 mph.In which direction is the plane pointed?
A)6.79° W of N
B)83.2° W of N
C)44.8° W of N
D)83.2° E of N
E)45.2° E of N

Accepted Answer

The plane must be pointed slightly west of north to counteract the eastward breeze and maintain a northerly path. The angle can be found using trigonometry, specifically the arctan function, with the eastward breeze as the opposite side and the plane's airspeed as the adjacent side of a right triangle. The correct calculation leads to an angle of approximately 6.79° west of north.

Question 6

A stone is thrown with an initial speed of 11.5 m/s at an angle of 50.0 above the horizontal from the top of a 30.0-m-tall building. Assume air resistance is negligible, and g = 9.8 m/s2.
-What is the magnitude of the horizontal displacement of the rock?
A)3.53 m
B)13.3 m
C)222 m
D)26.1 m

Accepted Answer

The horizontal displacement of a projectile can be found using the formula $d = v \cos(\theta) \times t$, where $v$ is the initial velocity, $\theta$ is the launch angle, and $t$ is the time of flight. To find $t$, we use the vertical motion equation $y = v \sin(\theta) t - \frac{1}{2} g t^2$, where $y$ is the vertical displacement. Since the stone is thrown from a height, we set $y = -30.0 \, \text{m}$ (taking downward as negative), $v = 11.5 \, \text{m/s}$, $\theta = 50^\circ$, and $g = 9.8 \, \text{m/s}^2$. Solving for $t$ gives the time of flight, which can then be used to calculate the horizontal displacement. The correct calculation leads to a displacement of approximately 26.1 meters, making option D the correct answer.

Question 7

A boat moves through the water in a river at a speed of 8 m/s relative to the water.The boat makes a trip downstream and then makes a return trip upstream to the original starting place.Which trip takes longer?
A)the downstream trip
B)the upstream trip
C)Both trips take the same amount of time.
D)The answer cannot be figured without knowing the speed of the river flow.

Accepted Answer

The upstream trip takes longer because the river's flow opposes the boat's motion, effectively reducing its speed relative to the ground.

Question 8

A jet airliner moving at 470 mph due east moves into a region where the wind is blowing at 110 mph in a direction 30.0° north of east.What is the new velocity and direction of the aircraft?
A)568 mph,5.56° N of E
B)573 mph,5.98° N of E
C)520 mph,9.06° N of E
D)568 mph,10.4° N of E
E)448 mph,9.45° N of E

Accepted Answer

The new velocity and direction can be found using vector addition. The eastward component of the wind's velocity is $110 \cos(30^\circ)$ mph and the northward component is $110 \sin(30^\circ)$ mph. Adding these to the plane's velocity gives a total eastward velocity of $470 + 110 \cos(30^\circ)$ mph and a northward velocity of $110 \sin(30^\circ)$ mph. The magnitude of the resultant velocity is $\sqrt{(470 + 110 \cos(30^\circ))^2 + (110 \sin(30^\circ))^2}$ mph, which calculates to approximately 568 mph. The direction, $\theta$, can be found using the tangent function: $\theta = \tan^{-1}\left(\frac{110 \sin(30^\circ)}{470 + 110 \cos(30^\circ)}\right)$, which gives about 5.56° north of east.

Question 9

A projectile is launched at an angle less than 30° and at speed v.Which will increase the projectile's range more,ignoring atmospheric drag?
A)doubling the launch angle
B)increasing the speed of launch by 50%
C)reducing the mass of the projectile
D)launching on February 29th

Accepted Answer

Increasing the speed of launch by 50% will significantly increase the range of the projectile. The range of a projectile is directly proportional to the square of its launch speed when launched at a fixed angle, making speed a critical factor in maximizing range. Choices A, C, and D do not directly influence the range in the same significant manner.

Question 10

Plane A is flying at 440 mph in the northeast direction relative to the Earth.Plane B is flying at 550 mph in the north direction relative to the Earth.What is the speed of Plane B as observed from Plane A?
A)570 mph
B)916 mph
C)392 mph
D)704 mph
E)714 mph

Accepted Answer

To find the speed of Plane B as observed from Plane A, we need to find the resultant velocity of Plane B relative to Plane A.

We can use vector addition to add the velocities of Plane A and Plane B.

Let's take a coordinate system with the x-axis pointing east and y-axis pointing north.

Velocity of Plane A = 440 mph at 45 degrees north of east 
= 440(cos 45 i + sin 45 j) mph

Velocity of Plane B = 550 mph at 90 degrees north 
= 550(j) mph

Adding the two velocities, we get:

Relative velocity of Plane B w.r.t. Plane A 
= (440cos45 + 550) j mph 
= 392 j mph

So, the speed of Plane B as observed from Plane A is 392 mph.

Therefore, the answer is (C).

Question 11

A boat moves at 10.8 m/s relative to the water.If the boat is in a river where the current is 2.00 m/s,how long does it take the boat to make a complete round trip of 1 100 m upstream followed by a 1 100-m trip downstream?
A)250 s
B)172 s
C)211 s
D)297 s
E)​336 s

Accepted Answer

The boat's effective speed upstream is 10.8 m/s - 2.0 m/s = 8.8 m/s, and downstream is 10.8 m/s + 2.0 m/s = 12.8 m/s. Time taken upstream = 1100 m / 8.8 m/s = 125 s, and downstream = 1100 m / 12.8 m/s = 85.94 s. Total time = 125 s + 85.94 s = 210.94 s, which rounds to 211 s.

Question 12

A plane is moving due north,directly towards its destination.Its airspeed is 220 mph.A constant breeze is blowing from west to east at 50 mph.How long will it take for the plane to travel 220 miles north?​
A)one hour
B)more than one hour
C)less than one hour
D)more information is needed
E)​none of the above

Accepted Answer

The plane's airspeed is given as 220 mph, which means it is not affected by the wind because the speed of the wind is perpendicular to the plane's direction. However, the wind affects the ground speed of the plane. The wind is blowing from west to east at 50 mph, which means it will decrease the ground speed of the plane. Therefore, the actual speed of the plane (ground speed) is less than the airspeed. We can calculate the ground speed of the plane by finding the vector difference between the airspeed of the plane (220 mph due north) and the velocity of the wind (50 mph east). Using Pythagoras' theorem, we have:

ground speed = sqrt(220^2 + 50^2) ≈ 226.27 mph

Now, we can calculate the time taken by the plane to travel 220 miles north using the formula:

time = distance/speed

time = 220/226.27 ≈ 0.97 hours, which is more than one hour. Therefore, the correct answer is B.

Question 13

A stone is thrown with an initial speed of 11.5 m/s at an angle of 50.0 above the horizontal from the top of a 30.0-m-tall building. Assume air resistance is negligible, and g = 9.8 m/s2.
-What is the magnitude of the horizontal-velocity component of the rock as it hits the ground?
A)79.5 m/s
B)8.81 m/s
C)58.7 m/s
D)7.39 m/s

Accepted Answer

The answer of A stone is thrown with an initial...

Question 14

A plane is moving due north,directly towards its destination.Its airspeed is 210 mph.A constant breeze is blowing from west to east at 20.0 mph.At what rate is the plane moving north?
A)209 mph
B)211 mph
C)204 mph
D)199 mph
E)194 mph

Accepted Answer

The plane's northward speed is slightly less than its airspeed due to the west-to-east breeze. Using the Pythagorean theorem, the northward speed can be calculated as the square root of (210^2 - 20^2) = 209 mph.

Question 15

A river flows due east at 2.0 m/s.A boat crosses the 300-m-wide river by maintaining a constant velocity of 15 m/s due north relative to the water.If no correction is made for the current,how far downstream does the boat move by the time it reaches the far shore?​
A)80 m
B)30 m
C)25 m
D)40 m
E)​50 m

Accepted Answer

We can use the formula:

distance = speed x time

To find the time it takes for the boat to cross the river, we can use:

time = distance / speed
time = 300 m / 15 m/s
time = 20 s

During this time, the river current will have pushed the boat downstream. The distance the boat moves downstream can be found using:

distance = speed x time
distance = 2 m/s x 20 s
distance = 40 m

Therefore, the boat moves 40 meters downstream by the time it reaches the far shore. The correct answer is D) 40 m.

Question 16

A stone is thrown with an initial speed of 11.5 m/s at an angle of 50.0 above the horizontal from the top of a 30.0-m-tall building. Assume air resistance is negligible, and g = 9.8 m/s2.
-What is the magnitude of the vertical-velocity component of the rock as it hits the ground?
A)8.81 m/s
B)7.39 m/s
C)25.8 m/s
D)19.3 m/s

Accepted Answer

The vertical-velocity component of the rock as it hits the ground can be found using the kinematic equation: $v_f = v_i + at$, where $v_f$ is the final velocity, $v_i$ is the initial vertical velocity, $a$ is the acceleration due to gravity, and $t$ is the time of flight. However, to directly calculate the final vertical velocity upon impact without explicitly finding the time of flight, we can use the energy conservation principle or the kinematic equation that relates initial velocity, final velocity, acceleration, and displacement: $v_f^2 = v_i^2 + 2ad$, where $d$ is the vertical displacement.Given:- Initial speed $v = 11.5$ m/s- Angle $\theta = 50^\circ$- Height of the building $h = 30.0$ m- Acceleration due to gravity $g = 9.8$ m/s$^2$The initial vertical velocity ($v_{iy}$) can be calculated as $v \sin(\theta)$:$v_{iy} = 11.5 \sin(50^\circ) \approx 8.81$ m/s (But this is not what we're solving for; it's just a step.)The vertical displacement from the top of the building to the ground is $d = -30.0$ m (negative because the displacement is downward).Using the equation $v_f^2 = v_i^2 + 2ad$ for the vertical component:$v_{fy}^2 = v_{iy}^2 + 2g(d)$Plugging in the values:$v_{fy}^2 = (8.81)^2 + 2(9.8)(-30.0)$Solving for $v_{fy}$ gives a negative value because the displacement $d$ is negative, indicating the direction of the velocity is downward. However, we are interested in the magnitude:$v_{fy} \approx 25.8$ m/sTherefore, the magnitude of the vertical-velocity component of the rock as it hits the ground is approximately 25.8 m/s.

Quiz 3: Motion in Two Dimensions

Plane A is flying at 350 mph in the northeast direction relative to the Earth.Plane B is flying at 460 mph in the north direction relative to the Earth.What is the direction of motion of Plane B as observed from Plane A?

A stone is thrown with an initial speed of 11.5 m/s at an angle of 50.0 above the horizontal from the top of a 30.0-m-tall building. Assume air resistance is negligible, and g = 9.8 m/s2. -What is the speed of the rock as it hits the ground?

A boat travels upstream and after one hour has gone 10 km.The boat next travels downstream and after one hour has gone 18 km.If the boat's speed relative to the water is constant,what is the speed of the current in the river?

A plane is moving due north,directly towards its destination.Its airspeed is 210 mph.A constant breeze is blowing from west to east at 25.0 mph.In which direction is the plane pointed?

A stone is thrown with an initial speed of 11.5 m/s at an angle of 50.0 above the horizontal from the top of a 30.0-m-tall building. Assume air resistance is negligible, and g = 9.8 m/s2. -What is the magnitude of the horizontal displacement of the rock?

A boat moves through the water in a river at a speed of 8 m/s relative to the water.The boat makes a trip downstream and then makes a return trip upstream to the original starting place.Which trip takes longer?

A jet airliner moving at 470 mph due east moves into a region where the wind is blowing at 110 mph in a direction 30.0° north of east.What is the new velocity and direction of the aircraft?

A projectile is launched at an angle less than 30° and at speed v.Which will increase the projectile's range more,ignoring atmospheric drag?

Plane A is flying at 440 mph in the northeast direction relative to the Earth.Plane B is flying at 550 mph in the north direction relative to the Earth.What is the speed of Plane B as observed from Plane A?

A boat moves at 10.8 m/s relative to the water.If the boat is in a river where the current is 2.00 m/s,how long does it take the boat to make a complete round trip of 1 100 m upstream followed by a 1 100-m trip downstream?

A plane is moving due north,directly towards its destination.Its airspeed is 220 mph.A constant breeze is blowing from west to east at 50 mph.How long will it take for the plane to travel 220 miles north?​

A stone is thrown with an initial speed of 11.5 m/s at an angle of 50.0 above the horizontal from the top of a 30.0-m-tall building. Assume air resistance is negligible, and g = 9.8 m/s2. -What is the magnitude of the horizontal-velocity component of the rock as it hits the ground?

A plane is moving due north,directly towards its destination.Its airspeed is 210 mph.A constant breeze is blowing from west to east at 20.0 mph.At what rate is the plane moving north?

A river flows due east at 2.0 m/s.A boat crosses the 300-m-wide river by maintaining a constant velocity of 15 m/s due north relative to the water.If no correction is made for the current,how far downstream does the boat move by the time it reaches the far shore?​

A stone is thrown with an initial speed of 11.5 m/s at an angle of 50.0 above the horizontal from the top of a 30.0-m-tall building. Assume air resistance is negligible, and g = 9.8 m/s2. -What is the magnitude of the vertical-velocity component of the rock as it hits the ground?

A plane is moving due north,directly towards its destination.Its airspeed is 220 mph.A constant breeze is blowing from west to east at 50 mph.How long will it take for the plane to travel 220 miles north?

A river flows due east at 2.0 m/s.A boat crosses the 300-m-wide river by maintaining a constant velocity of 15 m/s due north relative to the water.If no correction is made for the current,how far downstream does the boat move by the time it reaches the far shore?